Complex dynamical analysis of the Frackiewicz quantum commons game with bounded rationality

Abstract In the classical commons game, shepherds’ complete rationality causes the Tragedy of the Commons (ToC). Based on bounded rationality, we develop a dynamic Frackiewicz quantum commons game model to explore quantum entanglement’s effects on grazing strategies and profits, analyze quantum equilibrium stability, and apply synchronization control. Our Contributions: (1) The stability of quantum equilibrium depends on the adjustment speed and the squeezing parameter: it remains stable at low adjustment speeds and when the squeezing parameter is close to its critical value (i.e., π 4 ), but exhibits bifurcation or chaos otherwise. (2) Enhanced entanglement expands the stability region, mitigates individual rationality-induced conflicts, and boosts collective profits. Within the Frackiewicz framework, the quantum model optimizes ToC outcomes: the number of sheep never exceeds the classical level, while profits rise with entanglement and never fall below the classical level. (3) Synchronization control ensures strategy evolution convergence to stabilize the system. Physically, entanglement strengthens strategy correlations, suppresses bounded rationality-induced chaos, and expands the stable parameter region, offering a quantum solution to ToC.